Transient Fokker-Planck Equation solved with SPH

In many engineering matters, systems are submitted to random excitations. Probabilistic theories
aim at describing the properties of a system by means of statistical properties such as probability
density function (pdf). For a deterministic system randomly excited, the evolution of its pdf
is commonly described with Fokker-Planck-Kolmogorov equation (FPK). The FPK equation is a
conservation equation of a hypothetical fluid, which represents physically the transport of probability. To solve this equation, Smoothed Particle Hydrodynamics (SPH) are used: the system is
modelled with a conservation equation for the system and a transport equation for each particle.
Numerical implementation shows the superiority of this method over many other mesh-based
methods: (i) the conservation of total probability in the state space is explicitly written, (ii) no
specific boundary conditions must be imposed if an adaptive smoothing length is chosen and if
particles are initially regularly spread out, (iii) the positivity of the pdf is ensured.
Furthermore, thanks to the moving particles, this method is adapted for a large kind of initial
conditions (quasi-deterministic or even discontinuous). The FPK equation can be solved without
any a priori knowledge of the stationary distribution; just a precise representation of the initial
distribution is required.